Use thin_tac to throw away assumptions that you don't need in order to prove your subgoal. Thin_tac needs a scheme for knowing, which assumption it should use. The first matching assumption will be deleted.
lemma “⟦P t0;∀t. P t⟧ ⟹ P t0” apply (thin_tac “∀x. P x”) goal (1 subgoal): 1. P t0 ⟹ P t0
forget all assumptions: use a always matching scheme as often as possible apply (thin_tac “?x”)+